Typically, tomographic methods such as single photon emission computed tomography (SPECT) and positron emission tomography (PET) rely on measurements of poor quality due to broad spatial response functions imposed by properties of the particular system. In general, tomographic systems are used to infer a spatial distribution or estimate of a spatial distribution of a selected property of an object within the object's interior as determined from measurements of emanations from or through the object. For example, gamma rays emanate from an object in PET systems. Tomographic measuring mechanisms typically use an external source and detector as in the case of CT, or internal induced emanations and external detectors as in the case of PET.
Often times these spatial distributions are represented in picture form for visual analysis. Therefore these spatial distributions are commonly referred to as images although they need not necessarily be pictorially represented (e.g., the values at various points may be analyzed by computer to determine biological parameters or other object properties).
When a pictorial representation is displayed, the effect of broad spatial response functions manifests as a visual blurring. When, instead, the spatial distribution of interest is used to form estimates of some biological parameter, the effect causes a kind of numerical inaccuracy known as a partial volume effect.
PET as a biomedical imaging modality, is unique in its ability to provide quantitative information regarding biological function in a living subject. PET is a technique in which the distribution of a radioactive tracer material, introduced into an object, is reconstructed from the gamma rays that are emitted from within the object as a result of decay. PET is a powerful tool for brain research since, through proper selection of tracer materials, it provides non-invasive measurements of brain function through variables such as metabolism and blood flow. PET is also effective in assessing perfusion and tissue viability and, therefore, is widely used in cardiology and oncology.
Unfortunately, its use has been hampered by the poor spatial resolution of the images produced, resulting primarily from the relatively large detectors used to acquire the tomographic measurements. The poor image quality obtained by PET results from a number of factors, the most serious of which are blur due to broad spatial response functions and quantum noise due to detector response characteristics and limitations on permissible radiation dose.
In PET, each radioactive decay event, taking place within the object, leads to the simultaneous emission of two gamma rays in nearly opposite directions. These gamma rays are then counted by a detector system typically including one or more circular rings comprising a plurality of adjacent detectors positioned about the object. The detectors are connected to electronic coincidence circuitry.
An event is assumed to have taken place when two detectors register gamma rays at approximately the same instant of time. Since it is known that the gamma rays travel at 180 degree angles from one another, simultaneous detection serves to provide information regarding the location of the parent event.
FIG. 1 generally illustrates such a PET detection system. The ring of detectors 10 surrounds the object 12. A series of adjacently positioned detectors 14 form the ring 10. A pair of detectors 14a and 14b form a region known as a detector tube 16. With this configuration, an emission event taking place outside of the detector tube 16 cannot generally produce a pair of gamma rays that strike both detectors 14a and 14b. If a point source of emitting materials were placed at point 18 and moved to the right along line 20, the probability of the detection, by detectors 14a and 14b, of one of its emissions would increase as the source approached the center 22 of the detector tube 16 and then decrease as the source proceeded toward the edge of region and beyond.
The probability density function for the detections along such a lateral cross section of the detector tube 16 generally resembles curve 24. Curve 26 represents the probability function for an adjacent tube. The set of probability values, obtainable in this manner throughout the active region of a detector pair, is described herein as a spatial response function. Ideally, the spatial response function should have a value of zero for points outside of the boundaries of the detector tube; in practice, crosstalk between adjacent detectors precludes this.
The effect of a spatial response function is to determine in what proportion the source distribution of the object contributes on average to the number of gamma rays counted by the corresponding detector pair during the measurement process. Mathematically, if the object source distribution and a spatial response function are represented discretely, then, assuming corrections have been made for effects such as attenuation, the average number of gamma ray emissions detected by the corresponding detector pair can be described approximately as a sum of the density of tracer material within discrete elements of the object, weighted by the value of the corresponding spatial response function at that point. Symbolically, the operation can be represented generally by ##EQU1## where the x.sub.m denote the values for the object elements, the h.sub.m represent the corresponding elements of the spatial response function, and d is the average number of gamma ray emissions detected within the integration time of the measurement.
While the physical considerations from which this relationship arises are unique to PET, it is recognized in the art that tomographic measurement system can generally be represented as linear systems. Therefore, the term spatial response function will refer to those linear weighting factors relevant to a selected tomographic application. Since, for computational purposes, it is generally necessary to use discrete representations of continuous functions, spatial response function will refer, herein, to a vector of values that will be described as spatial response function elements.
Since the spatial response functions need not necessarily act mathematically on the object elements, and since the object property being measured varies with the particular application, the term signal will be used to refer generally to the true value of the quantity, of interest in a given situation, which is subject to linear transformation by the spatial response function or functions. For computational purposes, the signal will be considered to be divided into discrete values described herein as tomographic signal elements.
In applications other than PET, the observable quantity will not typically be a count of gamma ray emissions. Therefore, the term tomographic measurement will be used to describe the observed quantity, taking the place of d in Equation (1).
Generally, if the spatial response functions limit system spatial resolution, then the sampling rate afforded by the fixed positions of the detectors is typically insufficient to adequately characterize a signal out to the spatial frequency bandwidth determined by the spatial response functions. One solution, commonly employed in PET, moves the detection system to various positions to enable additional samples to be obtained. Using conventional methods, these extra samples in principle allow the signal to be specified accurately up to the bandwidth associated with the detector response functions, but detector response remains the limit on achievable resolution.
Referring again to FIG. 1, when in a conventional PET system there occurs a simultaneous detection of two gamma rays by two detectors at different points in the ring, a decay event is assumed to have taken place somewhere along a line 28 connecting the centers of the two detectors 14a and 14b. The mathematical foundation for traditional tomographic reconstruction techniques assumes the available measurements to consist of integrals of some object property along idealized rays of infinitesimal width, known as projection lines. In the conventional method for reconstructing PET images, the lines connecting the centers of the detectors take the place of these idealized projection lines. This is a very poor approximation which contributes to the low quality of conventional PET images. In reality, because actual detectors have finite extent, the simultaneous detection of two gamma rays proves that a decay event has taken place, not along an idealized ray, but within a broad region defined by the corresponding spatial response function.
Motion has been incorporated in many PET systems to improve sampling. One type of motion is commonly known as wobble. Wobble refers to an in-plane orbital motion of the entire detection system without rotation.
Referring to FIG. 2, when viewed edge-on, the circular wobbling 30 of the detector ring 10 manifests as a one-dimensional simple-harmonic motion. Hence, in the context of a single angular view in the parallel-beam geometry, the projection lines 28 (detector tubes) sweep back and forth with a sinusoidal time dependence.
In the usual procedure, the detection system travels in a continuous motion, and gamma rays are counted (measured) throughout. The gamma rays counted during time intervals of equal duration are grouped together and are considered as single observations 32 constituting additional samples of the signal along the direction of the projection profile. The gamma rays detected during a single time interval are usually treated as having been observed, not over a range of positions as is actually the case, but at a single point within that range. It is possible to stop the motion at discrete positions to achieve precisely what is approximated by the continuous motion, however, for practical reasons of mechanical design, continuous motion is more commonly employed.
The traditional approach to processing raw tomographic measurements 32 from such a system is to assign the counts observed during the motion to a plurality of equally spaced points 34 along an axis 36 of a projection profile by traditional interpolation methods such as bilinear interpolation. In tomographic imaging applications, this step, known as rebinning, yields a projection matrix (or sinogram). After rebinning, corrections are made to the projection matrix to compensate for such factors as attenuation and scattering of gamma rays through tissue in the brain, and/or system variations such as variation in detector sensitivity and other necessary corrections.
The corrected projection matrix is then used to reconstruct the image using various techniques known in the art. The rebinning step is complicated by the fact that the measurements (samples) are distributed in a highly non-uniform way due to the ring geometry of the data-acquisition system and the nature of the wobble motion.
The bilinear interpolation method typically begins with computation of the radial position and angular orientation of each projection line 28. The gamma-ray counts attributed, in the conventional processing approach, to a particular idealized projection line 38 are divided between two grid points 40 and 42 along the axis of the projection profile 36. The two grid points 40 and 42 neighbor an actual coordinate 44 corresponding to the particular idealized projection line 38. The division of the counts is made in proportion to the relative distances of the grid points 40 and 42 to the actual coordinate 44. For example, if the neighboring grid points 40 and 42 are separated by a distance .DELTA.r, and if the actual coordinate 44 lines at a distance s from A, then the fraction s/.DELTA.r of the counts associated with the true coordinate is assigned to point B and the remaining fraction is assigned to point A.
A blurring problem occurs using this approach since regardless of the number of samples obtained by the moving detection system, the spatial resolution of the signal cannot be improved beyond the limit imposed by the spatial response functions. Such conventional processing techniques used in the practice of PET and other tomography systems typically fail to take into account the broad spatial response functions that degrade the signal. Instead conventional processing techniques assume idealized projection lines and assume infinitesimal detector elements.
Non-moving tomographic systems are known that have improved imaging over conventional moving systems but typically must use a large number of smaller and more expensive detectors to acquire an image. Such non-moving systems that use interpolation methods generally suffer from the same effects of blurring as do conventional moving systems. Also, improving existing moving systems by incorporating the smaller detectors is generally cost prohibitive given the high cost of smaller detectors and the cost of modifying existing hardware. There exists a need for a signal recovery method that improves signals and is compatible with designs for moving systems that use less expensive, larger detectors to help eliminate the need for higher cost detectors while improving the quality of the tomographic images.
Therefore, there exists a need for a method of reducing the effect of blur due to spatial response function in tomographic detection systems. Furthermore, there exists a need for a method of recovering signals in tomographic systems that takes into account effects of spatial response functions rather than assuming idealized conditions. A need also exists for an improved signal recovery method which substantially reduces the computational time for recovering tomographics signals.